Rayleigh similarity solutions and boundary layer flow for concentrated suspensions

TLDR: In this paper, a series of transient problems in the flows of concentrated suspensions were investigated to test the effects of particle migration on the evolution of concentration and velocity profiles, and the authors reported a similarity solution to a Rayleigh problem, where the boundary of the infinite half space is given a velocity proportional to the square root of time.

Abstract: We have investigated a series of transient problems in the flows of concentrated suspensions to test the effects of particle migration on the evolution of concentration and velocity profiles. First, we report a similarity solution to a Rayleigh problem, where the boundary of the infinite half space is given a velocity proportional to the square root of time. Next, the classical Rayleigh problem, where the boundary is impulsively started initially at a constant velocity, is examined. The structure of the kinematics resembles that obtained in the first problem, but the concentration does not have a similarity form, and tends asymptotically to a uniform profile at large time. Finally, we solve the flow of a suspension past a semi-infinite plate, and discuss its connection to the Rayleigh problem. In all three cases, our calculations reveal Newtonian kinematics in the practical limit of a L ⪡ 1 , where a is the particle size, and L is a viscous diffusion length scale. In addition we see vastly different time and length scales in the evolution of the velocity and the concentration profiles. The velocity develops faster in time (by O( a L ) 2 ), and extends further in space (by O( L a ) ) than the concentration profile.